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Theorem bnj927 29576
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj927.1  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj927.2  |-  C  e. 
_V
Assertion
Ref Expression
bnj927  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )

Proof of Theorem bnj927
StepHypRef Expression
1 simpr 462 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  f  Fn  n )
2 vex 3084 . . . . . 6  |-  n  e. 
_V
3 bnj927.2 . . . . . 6  |-  C  e. 
_V
42, 3fnsn 5651 . . . . 5  |-  { <. n ,  C >. }  Fn  { n }
54a1i 11 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  { <. n ,  C >. }  Fn  {
n } )
6 bnj521 29541 . . . . 5  |-  ( n  i^i  { n }
)  =  (/)
76a1i 11 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( n  i^i  { n } )  =  (/) )
8 fnun 5697 . . . 4  |-  ( ( ( f  Fn  n  /\  { <. n ,  C >. }  Fn  { n } )  /\  (
n  i^i  { n } )  =  (/) )  ->  ( f  u. 
{ <. n ,  C >. } )  Fn  (
n  u.  { n } ) )
91, 5, 7, 8syl21anc 1263 . . 3  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( f  u.  { <. n ,  C >. } )  Fn  (
n  u.  { n } ) )
10 bnj927.1 . . . 4  |-  G  =  ( f  u.  { <. n ,  C >. } )
1110fneq1i 5685 . . 3  |-  ( G  Fn  ( n  u. 
{ n } )  <-> 
( f  u.  { <. n ,  C >. } )  Fn  ( n  u.  { n }
) )
129, 11sylibr 215 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  ( n  u.  { n } ) )
13 df-suc 5445 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2440 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
1514biimpi 197 . . . 4  |-  ( p  =  suc  n  ->  p  =  ( n  u.  { n } ) )
1615adantr 466 . . 3  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  p  =  ( n  u.  { n } ) )
1716fneq2d 5682 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( G  Fn  p  <->  G  Fn  (
n  u.  { n } ) ) )
1812, 17mpbird 235 1  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    u. cun 3434    i^i cin 3435   (/)c0 3761   {csn 3996   <.cop 4002   suc csuc 5441    Fn wfn 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657  ax-reg 8110
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-suc 5445  df-fun 5600  df-fn 5601
This theorem is referenced by:  bnj941  29580  bnj929  29743
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